The two answers so far are excellent. I'd just like to point out that "Would you like" can also be used as an ordinary conditional as in, "If it stops raining, would you like to go for a walk?"
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David RicherbyJul 24 '14 at 9:18

1

To me this isn't a true conditional, but rather a softer version of "do you want to go for a walk?" A true conditional would be "If it stops raining, would you consider going for a walk? -or- ...would you be up for a walk? However "would you like" is a mitigated form used for proposing or offering.
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CocoPopJul 24 '14 at 17:07

Do you like is used to ask if someone generally enjoys or is partial to something.

Would you like is a politer way of asking "do you want" when offering something. In English, as in many languages, the verb "want" is considered very direct, and the conditional serves to soften it a bit.

The other answers are not incorrect. They are technically correct, which is the best kind of correct.

However, there are cases that can blur the lines between the two. Consider the following instances:

Bob: Man, I'm starving.

Alice: Do you like pizza?

In this case, Alice is asking "do you like" but because Bob has already said he's looking to have some food, it is implied that Alice has some pizza to alleviate Bob's hunger.

This can also be unsolicited, such as

Alice grabs plate of nachos and motions them towards Bob

Alice: Do you like nachos?

Here Alice still has something to offer Bob, using "Do you like". There's a number of replies. A simple "no" would be inappropriate: Bob should indicate whether he is stating he doesn't like nachos, or whether he just doesn't want any right now.

These are both technically incorrect on Alice's part. (Get it together, Alice!) However, they're common enough that I wouldn't consider them fringe cases.

In the former case, for Alice to say "would you like some pizza" might be misconstrued as an immediate offer of food; in the latter case, it might be misconstrued as implying that Alice prepared the nachos in the expectation that Bob liked them. Both offer easy paths for the conversation that avoid awkwardness whether or not Bob likes nachos.
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supercatJul 25 '14 at 0:00

The difference is that these phrases are using a lot of subtext. They're actually violating the "correct" use of the phrases, but doing so deliberately to convey/imply extra non-verbal information. This is the sort of thing that would confuse a ELL so good to know about, but is probably not helpful for understanding the difference between the phrases and when they should be used.
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SevenSidedDieJul 25 '14 at 14:42

In the end, though, it doesn't matter. Bob got tired of Alice always sending him encrypted letters and they have since broke up.
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corsiKaDec 31 '14 at 21:58